Adult Link's Trading Sequence

Note that you'll need Epona to get through this trading sequence. Also remember you have to actually show them the item using a C-button, not just talk to them.

The egg
Talk to the chicken woman in Kakariko Village to get the mini chicken egg.

The chicken
Wait for a day for the egg to hatch (or just play Song of the Sun twice).

The blue chicken
Use the mini-chicken to wake Talon in the house on the ground level of Kakariko village. Bring it back to the chicken woman to get the blue chicken.

The odd mushroom
Find a sleeping man in the lost woods (just go left from the entrance) and wake him with the blue chicken. He'll give you an odd mushroom.

The odd potion
Take the odd mushroom to the weird potion shop in Kakariko Village (Go to the normal potion shop and go through the hallway on the left to get to the fenced off section of the village, then go up a ladder to get to the weird potion shop.) before the time runs out. If you try using your ocarina to warp it will instantly spoil. A good shortcut is if you've planted the magic bean as young Link in the soft soil one screen to the left of the sleeping man, you can ride the bean platform up onto the bridge and then exit to get out of Kokiri Forest quickly (it's the bridge that leads out). Give the shop owner the mushroom and you'll get an odd potion.

Poacher's Saw
Find the small girl in the lost woods (She's where the sleeping man was, left of the entrance) and trade the odd potion for the Poacher's Saw

Broken Goron's Sword
Give the Poacher's Saw to the carpenter's boss in Gerudo Valley to get the broken Goron's Sword. You can get over the broken bridge by jumping with Epona or using a longshot.

Prescription
Give the broken Goron's Sword to the Big Goron at the summit of Death Mountain (NOT the one inside Goron City) and get the Prescription. Climb to the top of Death Mountain and then look at the right edge of the plateau up there to find the Big Goron.

Eyeball Frog
Bring the prescription to King Zora to get the Eyeball Frog. Remember to thaw him first by using magic blue fire, which you can get from the potion shop in Kakariko Village for 300 rupees, or from the Ice Cavern .

Eye drops
Bring the Eyeball Frog to the professor in the Lake Hylia lab before the time runs out to get the Eye drops. If you try using your Ocarina to warp it will spoil instantly. The quickest route is to take a left as soon as you exit Zora's Domain and dive in the small pool there to warp to the Lost Woods. As soon as you pop up in the woods hang a left and you'll warp to Kokiri Forest. Then exit the forest, summon Epona with Epona's Song, and ride quickly South to Lake Hylia. Jump the fences blocking the entrance to Lake Hylia and ride Epona right up the the lab's front door.

Claim Check
Bring the Eye drops back to the Big Goron at the summit of Death Mountain before the time runs out to get the Claim Check. A quicker way to get to him is to ride Epona to the Lost Woods, go R,L,F to warp into the Goron City, then exit (go left, take the second passageway on your left, up the stairs, then right and take the first passageway on your right) and climb up to the Big Goron. When you get to the part with the falling rocks (if you haven't beaten Fire Temple yet) just roll attack with forward + A constantly to avoid most of them. You'll have to use your hookshot or arrows to kill the Skulltulas on the wall unless you're good at dodging them.

Big Goron's Sword
Wait for three days and return to the Big Goron at the summit of Death Mountain to get the big Goron's Sword. You can play Song of the Sun 6 times if you're impatient.